Solve:
$5 + 5^{2} + 5^{3} + . . . . . . + 5^{n}$

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Solution

$5 + 5^{2} + 5^{3} + . . . . . + 5^{n}$
The above series forms G.P. with first term $5$ and common ratio $5$ .
No. of terms $= n$
As we know that sum of $n$ terms in a G.P. is given as-
$S_{n} = \frac{a (r^{n} - 1)}{r - 1}; r > 1$
Therefore, for the given series,
$S_{n} = \frac{5 (5^{n} - 1)}{5 - 1} = \frac{5}{4} (5^{n} - 1)$

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